The asymptotics of the partition of the cube into Weyl simplices, and an encoding of a Bernoulli scheme∗ A. M. Vershiky 02.02.2019 Abstract We suggest a combinatorial method of encoding continuous symbolic dynamical systems. A con- tinuous phase space, the infinite-dimensional cube, turns into the path space of a tree, and the shift is mapped to a transformation which was called a \transfer." The central problem is that of dis- tinguishability: does the encoding separate almost all points of the space? The main result says that the partition of the cube into Weyl simplices satisfies this property.1 Contents 1 Introduction 2 2 Classical and combinatorial encoding of transformations 4 2.1 Classical encoding and generators . 4 2.2 Combinatorial encoding . 4 2.3 The frame as a combinatorial invariant of an encoding . 5 2.4 Distinguishability problem . 6 2.5 Entropy estimates . 8 3 The main example: encoding a Bernoulli scheme by Weyl simplices and the trian- gular compactum M 9 3.1 The partition of the cube into Weyl simplices . 9 3.2 The triangular compactum of paths in the tree of i-permutations . 11 arXiv:1904.02924v1 [math.CO] 5 Apr 2019 3.3 An isomorphism between the cube I1 and the triangular compactum. The positive answer to the distinguishability problem . 12 3.4 Transfer for the tree of i-permutations and the triangular compactum . 14 ∗Partially supported by the RFBR grant 17-01-00433. ySt. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg State University, and Institute for Information Transmission Problems. 1Keywords: combinatorial encoding, transfer, Bernoulli scheme, graded graph. 1 4 Comments and remarks 15 4.1 More on the distinguishability problem . 16 4.2 The list of isomorphisms . 16 4.3 The relation to matrix distributions . 16 4.4 Other compactifications of symmetric groups . 17 4.5 Relation to C∗-algebras . 17 4.6 Representations of the infinite symmetric group . 18 4.7 An analog of the partition into Weyl simplices for an arbitrary Bernoulli scheme . 19 4.8 Distinguishability of random matrices . 19 5 Graded graphs, transfer, and quasi-stationary processes 19 5.1 Why not only trees . 19 5.2 Definition of transfer for a graded graph . 20 1 Introduction In the theory of dynamical systems and, more generally, measure theory, the following question often proves fruitful: find a cover of a given measure space by a space endowed with a product measure (i.e., a product of independent variables) for which a given transformation (or a group of transformations) is a homomorphic image of the shift (respectively, are homomorphic images of the shifts) in the covering space. If such a cover (homomorphism) exists, and even is an isomorphism, then we obtain important information on the original object. On the other hand, the inverse problem is not less important: how one can economically encode a sequence of independent identically distributed continuous random variables (i.e., a Bernoulli scheme) using finite codes? That is, can one replace a continuous scheme by a locally finite one and how can this be done? In both cases, it is important to find a method of economical encoding of a continuous scheme and Bernoulli shift, or even a more general system. We suggest a general method of combinatorial encoding of a Bernoulli scheme, and consider a sim- plest nontrivial example of such an encoding, using the partition of the cube into Weyl simplices. This example is related to a simplest tree (of permutations) and produces an isomorphism between the classical Bernoulli scheme with a continuous set of states and a new type of a measure-preserving transformation called transfer. This is a nonstationary Markov shift acting in the Cantor-like space of paths in a graded graph (in the simplest case, in the path space of a tree). In this context, objects of ergodic theory become related to the combinatorial theory of graded graphs and, consequently, to combinatorics and representation theory. The main problem arising here is the distinguishability problem: is the encoding faithful, i.e., does it separate almost all points of the Bernoulli scheme? In other words, is the encoded system isomorphic to the original one or is it only a homomorphic image of this system? In the example with Weyl simplices, this problem can be stated very simply: can one recover a realization of a Bernoulli scheme with state space [0; 1] from all pairwise inequalities between its coordinates? Quite paradoxically, the 2 answer (see Section 3) is positive: this can be done with probability 1. Actually, this question is related to the theory of equidistributed sequences, but the extensive literature on the subject seems to contain no mention of this fact. A much more complicated example is related to the Young graph and the RSK correspondence. It was initiated by the old paper [3] and was resolved, also positively, in the recent papers [5, 6]. In this case, the encoding uses Q-tableaux of the RSK correspondence. The proofs in [5, 6] are based on a thorough analysis of the theorem on the limit shape of Young diagrams and the study of the Sch¨utzenberger transformation. We will return to this example in another paper, where we will apply a general method of resolv- ing the distinguishability problem, which consists in proving certain laws of large numbers \along" realizations. These laws can be quite complicated. For example, our proof is based on a new theorem on the limit shape for P -tableaux; however, the approach itself, which is mentioned in the present paper too, is universal. It is worth noting that our notion of transfer (see Sections 4 and 5) is a far generalization of the Sch¨utzenberger transformation (jeu de taquin) and seems to be important for the general theory of graphs and transformations with an invariant measure. Describing all invariant measures for a transfer and the study of its properties is a new interesting area of the theory of dynamical systems and the combinatorics of graphs. The main technique here is a combinatorial method of studying increasing invariant sequences of finite measurable partitions of Lebesgue spaces, which are, according to V. A. Rokhlin, separable complete measure spaces, or, in other words, spaces isomorphic mod 0 to an interval with the Lebesgue measure (if the original measure is continuous). In the author's opinion, the most important problems in measure theory and its applications to different fields of mathematics are related to the geometry and combinatorics of σ-algebras (= measurable partitions) and their sequences. Here we are interested in infinite increasing sequences; the combinatorics underlying the theory of such sequences is the study of properties of infinite trees and graded graphs. Another class of sequences of σ-algebras is that of decreasing sequences, or filtrations (see [9]); the corresponding theories are closely related, but strongly different. In Section 2, we define a combinatorial encoding and state the main problems. In particular, we define a frame, i.e., a tree endowed with a translation which is a combinatorial invariant of an in- creasing sequence of partitions. We state and discuss the distinguishability problem, and also discuss numerical invariants of exhausting sequences, for which the distinguishability problem has a positive answer. The main Section 3 introduces an encoding of a Bernoulli scheme by Weyl simplices (= in- tersections of Weyl chambers with the unit cube); it is used to establish an isomorphism between a continuous Bernoulli shift and the transfer of a triangular compactum (= the path space of the tree of i-permutations). Section 4 contains various problems related to this example and its generalizations. In particular, we state a problem on different compactifications of the infinite symmetric group, one being the compactum of virtual permutations and another one being the main example of this paper. In Section 5, we give a general definition of transfer for a graded graph and describe problems related to this notion. 3 2 Classical and combinatorial encoding of transformations 2.1 Classical encoding and generators We begin by recalling the method of encoding endomorphisms and automorphisms or, more generally, arbitrary actions of groups and semigroups with an invariant measure used in symbolic dynamics, when one defines an action of the group by shifts in the space of functions on the group with a shift-invariant measure. For simplicity, we consider a Bernoulli endomorphism (= one-sided Bernoulli shift) S, which is Q 1 1 a transformation of the space n I ≡ I , where I = [0; 1], with an invariant product measure m , where m is the Lebesgue measure on [0; 1]. The usual method of encoding endomorphisms (and automorphisms) in ergodic theory and information theory is to choose an I-valued measurable function f : I1 ! I on the space of trajectories I1 of the process (\symbolic space") and study the family of −n 1 1 1 all its shifts ff(S ·)gn=1. If these shifts (regarded as functions on I ) separate the points of I , i.e., the product of the partitions of I1 into the preimages of points corresponding to the shifts of f is the partition into singletons, then the partition into the level sets of the original function (and the function itself) is called a generator. In this case, we obtain a new isomorphic model of the shift in the same space I1, but, in general, with another invariant measure different from m1. To check whether a given function is a generator is a difficult and instructive problem even for a Bernoulli scheme, and more so in the general case. But since the product of partitions is a partition invariant under the shift, we always have a well-defined quotient endomorphism, which may or may not be isomorphic to the original endomorphism.